Let ${\displaystyle (X_{t})_{t\in T}}$  be a stochastic process. In most cases, ${\displaystyle T=\mathbb {N} }$  or ${\displaystyle T=\mathbb {R} ^{+}}$ . Then the stochastic process has independent increments if and only if for every ${\displaystyle m\in \mathbb {N} }$  and any choice ${\displaystyle t_{0},t_{1},t_{2},\dots ,t_{m-1},t_{m}\in T}$  with

${\displaystyle t_{0}<t_{1}<t_{2}<\dots <t_{m}}$ 

the random variables

${\displaystyle (X_{t_{1}}-X_{t_{0}}),(X_{t_{2}}-X_{t_{1}}),\dots ,(X_{t_{m}}-X_{t_{m-1}})}$ 

are stochastically independent.^[2]

A random measure ${\displaystyle \xi }$  has got independent increments if and only if the random variables ${\displaystyle \xi (B_{1}),\xi (B_{2}),\dots ,\xi (B_{m})}$  are stochastically independent for every selection of pairwise disjoint measurable sets ${\displaystyle B_{1},B_{2},\dots ,B_{m}}$  and every ${\displaystyle m\in \mathbb {N} }$ . ^[3]

Let ${\displaystyle \xi }$  be a random measure on ${\displaystyle S\times T}$  and define for every bounded measurable set ${\displaystyle B}$  the random measure ${\displaystyle \xi _{B}}$  on ${\displaystyle T}$  as

${\displaystyle \xi _{B}(\cdot ):=\xi (B\times \cdot )}$ 

Then ${\displaystyle \xi }$  is called a random measure with independent S-increments, if for all bounded sets ${\displaystyle B_{1},B_{2},\dots ,B_{n}}$  and all ${\displaystyle n\in \mathbb {N} }$  the random measures ${\displaystyle \xi _{B_{1}},\xi _{B_{2}},\dots ,\xi _{B_{n}}}$  are independent.^[4]

Independent increments are a basic property of many stochastic processes and are often incorporated in their definition. The notion of independent increments and independent S-increments of random measures plays an important role in the characterization of Poisson point process and infinite divisibility